S OPEN SITS 33

For every 2 1 , A, <i(.\\ ;*) -* </(A; .r) -f c/(j% 2), hence
</(A\ A) •. infc/U, 2) i, infWtv, r) 4 r/(j\ r)) *<•• </(A\.r) 4* inf </0\ 2)

/ if A

hy (2.3.8) and (2.3.10). Similarly one has </0\ A) < i/(,v,,r) 4 rf(A\ A),

For any nonempty net A in l\ the Jiewwtrr of A is defined as #(
sup </(v, r); Ills it positive real number or I <*>;Ac B implies #(A) «£ /KB),

* « \t v»? A

The relation $f A) ^ 0 hokls if and only if A is a one point Bet*

(3,43) Iw nwr /w//, rf(B'(«i; r)) * 2r,

I;or if (fat *) *•. r iiiiii f/(fi, r) * r, I/(A(, ,r) *; 2r hy the triangle inequality,

A fwumh'd wt in f- i* a nonempty net whose diameter h finite. Any hall
in bountitx), The whole space 1; can IH* boumleti, as the example of the extended
real line It shows. Any nonempty Mib\et of a bounded set is bounded.

(3,4,4) Tlw <*f run IwwttM ^»/,v A, B i,t

l;or if ii* A, /M II, tticnjf A»r are any two points in A «• ? B, either ,v
and r are in A, and then */( i* il * ^(A)» or they ate in ft fl|v» rl *"' rf(B)»
or for instaiur i * A and r# II. uml then il(u rl ^i' il{,if n| I rA«iv/>) t r/(/\ v)
hy thf triangle imH|uality, hci

*i| A* * II) - t tt{A» t

this If nc fitf any «* * A, A * fl» we4 have

i^A* ' III * «AA, III i c^lA) t

hy *tf *k\* W>

It tollowH that if A is binuuknl, IV »r any iw » I1* \ is in the

ball iif ,%„ and tadiits ill ifl , Ai I- «M A),

f.

In ti f, i/, 4ii ,ii*l i% a A of !*

Im % *, > ll|>, r M' A,

The set is t*ec U); the I' i%oes not belong to a ball B(a; r) (resp. B'(#; r)), then